analysisoftakeovertimeandconvergencerate … bany mohammed is an assistant professor at the mis...

Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 4, No. 4, 2013 305

Analysis of takeover time and convergence ratefor harmony search with novel selection methods

Iyad Abu Doush*Department of Computer Sciences,Information Technology College,Yarmouk University,21163, Irbid, JordanE-mail: [email protected]*Corresponding author

Mohammed Azmi Al-BetarSchool of Computer Sciences,Universiti Sains Malaysia,11800 USM, Penang, MalaysiaandDepartment of Computer Sciences,Alhuson College,50, Irbid, JordanE-mail: [email protected]

Ahamad Tajudin Khader andMohammed A. AwadallahSchool of Computer Sciences,Universiti Sains Malaysia,11800 USM, Penang, MalaysiaE-mail: [email protected]: mohd [email protected]

Ashraf Bany MohammedManagement Information Systems Department,University of Hail,P.O. Box 2440Hail 81451, Saudia ArabiaE-mail: [email protected]

Abstract: Recently, common selection schemes used in harmony searchalgorithm (HSA) are altered in memory consideration operation to imitate thenatural selection principle of survival of the fittest. The selection schemesadopted include: random, proportional, tournament, and linear rank. In thispaper, these selection schemes are analysed in order to evaluate their effecton the performance of HSA. The analysis considers takeover time and

Copyright © 2013 Inderscience Enterprises Ltd.

306 I.A. Doush et al.

convergence rate to measure the effectiveness of each selection scheme.Furthermore, a scaled proportional selection scheme is proposed to replacethe proportional selection scheme to overcome its shortcoming with negativefitness values. To study the effect of these different selection schemes weuse eight global optimisation functions with different characteristics. Anexperimental evaluation show that linear rank selection provides the highestconvergence speed and highest takeover time. On the other hand, scaledproportional selection provides the slowest convergence speed and slowesttakeover time. This indicates the effect of the type of the selection methodused in memory consideration in takeover time and convergence rate.

Keywords: harmony search algorithm; HSA; evolutionary algorithms; EA;selection mechanisms; meta-heuristic algorithm.

Reference to this paper should be made as follows: Doush, I.A.,Al-Betar, M.A., Khader, A.T., Awadallah, M.A. and Mohammed, A.B. (2013)‘Analysis of takeover time and convergence rate for harmony search withnovel selection methods’, Int. J. Mathematical Modelling and NumericalOptimisation, Vol. 4, No. 4, pp.305–322.

Biographical notes: Iyad Abu Doush is Assistant Professor and DepartmentHead at Computer Science Department in Yarmouk University, Irbid-Jordan.He received his BSc from Computer Science Department at YarmoukUniversity, Irbid-Jordan in 2001. In 2001, he was awarded a scholarship fromYarmouk University to study MSc in Computer Science and information, andin 2005 he was awarded a scholarship from Yarmouk University to studyPhD in Computer Science. He received his PhD from the Computer ScienceDepartment at New Mexico State University (NMSU), Las Cruces, USAin 2009. In 2009, he awarded Merit-based Enhancement Fellowship fromgraduate school for outstanding service to New Mexico State University asteaching and research assistant. He has published a number of high qualitypapers in international journals and conferences, and he is the reviewerfor several international conferences and journals. His research interests aremainly directed to web accessibility, multimodal interfaces, metaheuristicoptimisation and virtual environments.

Mohammed Azmi Al-Betar is an Assistant Professor at Department ofComputer Science in Al-zaytoonah University, Amman-Jordan and a seniormember of Computational Intelligence Research Group in the School ofComputer Science at University Sains Malaysia (USM). He received hisBSc and MSc from Computer Science Department at Yarmouk University,Irbid-Jordan in 2001 and 2003, respectively. He received his PhD from theSchool of Computer Sciences at University Sains Malaysia (USM), PulauPenang, Malaysia in 2010. He has published a number of high qualitypapers in international journals and conferences. His research interests aremainly directed to metaheuristic optimisation methods and hard combinatorialoptimisation problems including scheduling and timetabling.

Ahamad Tajudin Khader is Professor and Deputy Dean of Graduate Studiesand Research in School of Computer Sciences at Universiti Sains Malaysia.He finished his studies in BSc and MSc at Ohio, USA, and his PhD atStrachclyde University, UK. His research interests include genetic algorithms,

Analysis of takeover time and convergence rate for harmony search 307

particularly in the areas of population modelling and parameter control. Heis also actively involved in research on evolutionary algorithms and problemsin scheduling, timetabling and planning.

Mohammed A. Awadallah is a PhD student in School of Computer Sciencesat University Sains Malaysia (USM), Pulau Penang, Malaysia. His researchinterests are mainly directed to metaheuristic optimisation methods and hardcombinatorial optimisation problems including scheduling and timetabling.

Ashraf Bany Mohammed is an Assistant Professor at the MIS Departmentin Petra University, Jordan. He received his PhD from Seoul NationalUniversity from the School of Industrial Engineering. Prior to his work inPetra University, he had lectured in various IT institutions including YarmoukUniversity and Al-Balqa Applied University. He published a number ofresearch papers and is serving as a reviewer and committee member for anumber of journals and conferences. His current research interests include:cloud and grid computing economics, business modelling, e-infrastructure,e-government, innovation, ICT for development, social networking, and webservices.

1 Introduction

The selection process in evolutionary algorithms (EA) is choosing the best individualsfrom the population more often than worse individuals to form the next generation(Back, 1996). Better individuals will be selected according to their fitness values basedon ‘survival of the fittest’ principle which introduced by Darwin’s theory (Back, 1996).

The selection process is an important phase in the EA as it affects on theconvergence speed and the convergence precision (Hancock, 1994). The convergenceprecision can be described as the ability of the algorithm search process to reach theglobal optimal.

A good selection technique leads to a fast convergence with high probability offinding a global optimum (Goldberg, 1989). Several researchers studied how selectionprocess can improve the performance of EA including takeover time and convergencerate (Motoki, 2002; Miller and Goldberg, 1995; Baker, 1985; Blickle and Thiele, 1997;Hancock, 1994; Goldberg and Deb, 1991; Back et al., 1991). This process still needfurther investigation in HSA as several selection schemes can be modified and adoptedto the behaviour of HSA to enhance the convergence properties.

The takeover time concept was introduced by Goldberg and Deb (1991). Itmeasures the selective pressure by counting the number of generations needed to fillthe population, with copies of the best solution obtained in the initial population.Convergence rate can be described as the speed of the algorithm progress towards theoptimal solution.

One of the recent succesfull EA, developed by Geem et al. (2001) and Leeand Geem (2005), is harmony search algorithm (HSA). The HSA mimic the musicimprovisation process. It has been used by several researchers to solve several types ofoptimisation problems from different fields such as: timetabling, flow shop scheduling,structural optimisation, multi-buyer multi-vendor supply chain problem, the complexbridge system optimisation problem, dynamic economic load dispatch, and lot-streaming


flow shop scheduling problem (Lee and Geem, 2005; Taleizadeh et al., 2011; Al-Betarand Khader, 2010; Al-Betar et al., 2010a, 2010b, 2010c; Wang et al., 2011, 2010; Zouet al., 2010a; Pandi and Panigrahi, 2011; Pan et al., 2011). Several other problems arediscussed in Ingram and Zhang (2009) and Geem (2008a).

Several variations and improvements of the HSA has been proposed recently toadjust the algorithm to fit a specific problem, improved HSA (Mahdavi et al., 2007;Wang and Huang, 2010), global-best HSA (Omran and Mahdavi, 2008), stochasticderivative HSA (Geem, 2008b), and particle swarm HSA (Geem, 2009; Zhao et al.,2011). The goal of these different versions of HSA is to enhance the convergenceproperties.

The HSA is an iterative algorithm works as follows:

1 Start with a population randomly generated and stored in the harmony memory(HM).

2 Iteratively generate a new solution through three operators,

a memory consideration, in this operator a value of a variable is selected fromany solution stored in the HM

b random consideration, in this operator a value of a variable is randomlygenerated from its available range

c pitch adjustment, in this operator a value of a variable is adjusted for localimprovement.

The new solution is evaluated according to the objective function and it replaces theworst solution in the HM. The HSA will iterate until a specific stop criteria is met(Al-Betar et al., 2010c; Geem et al., 2001).

In the memory consideration operation, a value for the variable is selected asfollows: select a random solution xi from the HM, then take the value of thevariable from that solution xi. This is a random selection with less consideration of‘survival of the fittest’ principle. Instead of this random process, several other powerfulselection mechanisms are suggested and borrowed from other EA for HSA (Al-Betaret al., 2011), to improve the process of memory consideration including: proportional,tournament, and linear rank. The objective was to investigate the usability of suchselection schemes in memory consideration, and to what extent it will affect theperformance of HSA.

In this paper, the convergence rate and the takeover time are analysed to studythe behaviour of several selection schemes. The results obtained are compared withthe random selection scheme proposed by Geem et al. (2001) and Lee and Geem(2005). Then we conduct experiments to investigate the behaviour of the selectionschemes on benchmark functions with different characteristics. A two selection measuresare investigated to study the performance of each selection scheme: takeover timeand convergence rate. The outcomes from this study can help in identifying whatselection scheme is most appropriate for such optimisation problem with specificcharacteristics.

The remainder of this paper is organised as follows: in Section 2 an overview ofHSA is presented. Section 3 demonstrates an analysis of selection schemes adopted


in harmony search in terms of convergence rate and takeover time. Finally, Section 4conclude the paper with suggestions for future directions.

2 Harmony search algorithm

The HSA is a population-based algorithm inspired by musical improvisation process(Geem et al., 2001), when musicians tries to find a better harmony practice afterpractice. This idea is mapped to optimisation problems, a set of decision variablesis assigned with values, iteration by iteration, seeking for a ‘good enough’ solutionas evaluated by an objective function. HSA has five main steps described as follows(Al-Betar et al., 2011):

1 Initialise the algorithm parameters. Normally, the optimisation problem is initiallymodelled as: min{f(x) | x ∈ X}, where f(x) is the objective function;x = {xi | i = 1, . . . , N} is the set of decision variables. X = {Xi | i = 1, . . . , N} isthe possible value range for each decision variable, where Xi ∈ [LBi, UBi], whereLBi and UBi is the lower and upper bound for the decision variable xi

respectively and N is the number of decision variables. The parameters of theHSA required to solve the optimisation problem are also specified in this step:

a the harmony memory consideration rate (HMCR), used in the improvingprocess to determine if the value of a decision variable is to be selected fromthe solutions stored in the HM

b the harmony memory size (HMS) is an n-dimension vector similar to thepopulation size in genetic algorithm (GA)

c the pitch adjustment rate (PAR), decides whether the decision variables are tobe modified to a neighbouring value

d the distance bandwidth (BW) determines the adjustment value in the pitchadjustment operator

e the number of improvisations (NI) corresponds to the number of iterations.

These parameters will be explained in more detail in the next steps. Note that theHMCR and PAR are the two parameters responsible for the improvisation process.

2 Initialise the HM. The HM is an augmented matrix of size N × HMS whichcontains sets of solution vectors determined by HMS (see (1)). In this step, thesevectors are randomly generated as follows: xj

i = LBi + (UBi − LBi)× U(0, 1),∀i = 1, 2, . . . , N and ∀j = 1, 2, . . . ,HMS, and U(0, 1) generate a uniform randomnumber between 0 and 1. The generated solutions are stored in the HM inascending order according to their objective function values.

HM =

x11 x1

2 · · · x1N

x21 x2

2 · · · x2N

...... · · ·

...xHMS1 xHMS

2 · · · xHMSN

. (1)


3 Improvise a new harmony. In this step, the HSA will generate (improvise) a newharmony vector from scratch, x′ = (x′

1, x′2, · · · , x′

N ), based on three operators:memory consideration, random consideration, and pitch adjustment.

• Memory consideration. In memory consideration, the value of the firstdecision variable x′

1 is randomly selected from the historical values,{x1

1, x21, . . . , x

HMS1 }, stored in HM vectors. Values of the other decision

variables, (x′2, x

′3, . . . , x

′N ), are sequentially selected in the same manner with

probability (w.p.) HMCR where HMCR ∈ (0, 1). It is worth noting that theselection scheme in memory consideration is random and that the naturalselection principle is not used (i.e., the value of decision variable is selectedfrom any solution using unguided selection scheme).

• Random consideration. Decision variables that are not assigned with valuesaccording to memory consideration are randomly assigned according to theirpossible range by random consideration with a probability of (1-HMCR) asfollows:

x′i ←

{x′i ∈ {x1

i , x2i , . . . , x

HMSi } w.p. HMCR

x′i ∈ Xi w.p. 1-HMCR.

• Pitch adjustment. Each decision variable x′i of a new harmony vector,

x′ = (x′1, x

′2, x

′3, . . . , x

′N ), that has been assigned a value by memory

considerations is pitch adjusted with the probability of PAR wherePAR ∈ (0, 1) as follows:

Pitch adjusting decision for x′i ←

{Yes w.p. PARNo w.p. 1-PAR.

If the pitch adjustment decision for x′i is Yes, the value of x′

i is modified toits neighbouring value as follows: x′

i = x′i ± U(0, 1)× BW

4 Update HM. If the new harmony vector, x′ = (x′1, x

′2, · · · , x′

N ), is better than theworst harmony xworst stored in HM in terms of the objective function value(i.e., xworst = xHMS in case HM is sorted), the new harmony vector is included tothe HM, and the worst harmony vector is excluded from the HM. This is a greedyselection scheme where the principle of natural selection is applied.

5 Check stopping criterion. Step 3 and step 4 of HSA are repeated until the stopcriterion (maximum NI) is met. This is specified by the NI parameter.

3 Analysis of takeover time and convergence rate for HSA with differentselection methods

This section provides an analysis of the selection methods used in memory considerationin terms of takeover time and convergence rate. The selection methods are:random, tournament, scaled proportional, and linear rank. Initially, eight benchmarkfunctions are used for the analysis process as shown in Table 1 and described in thenext sections.


Table 1 Benchmark functions used to evaluate HSA variationsFu

nctio

nname

Expression

Search

range

Optimum

value

Category(Pan

etal.,2010)

Sphere

function

f1(x

)=

N ∑ i=1

x2 i

xi∈

min(f

1)

Unimodal

(OmranandMahdavi,2008)

[−100,1

00]

=f(0,...,0

)=

0

Step

function

f2(x

)=

N ∑ i=1

(⌊xi+

0.5⌋)

2xi∈

min(f

2)

Discontinuous


[−100,1

00]

=f(0,...,0

)=

0unimodal

Schw

efel’sproblem

2.26

f3(x

)=

−N ∑ i=1

( xisin(√ |x

i|))

xi∈

min(f

3)=

f(420.9687,...,

Difficult

(Yao

etal.,1999)

[−500,5

00]

420.9687)=

−12569.5

multim

odal

Six-hump

f4(x

)=

4x2 1−

2.1x4 1

xi∈

min(f

4)=

Low

camel-backfunction

+1 3x6 1+

x1x2

[−5,5

]f(−

0.08983,0.7126)

dimensional


−4x2 2+

4x4 2

=−1.0316285

Shifted

sphere

f5(x

)=

N ∑ i=1

z2 i+

fbias 1

xi∈

min(f

5)

Unimodal,shifted,

function

where

z=

x−

o[−

100,1

00]

=f(o

1,...,o

N)

separable,

(Suganthan

etal.,2005)

=fbias 1

=−450

andscalable

Shifted

Schw

efel’s

f6(x

)=

N ∑ i=1

( i ∑ j=1

z j

) 2xi∈

min(f

6)

Unimodal,shifted

problem

1.2

+fbias 2,

[−100,1

00]

=f(o

1,...,o

N)

non-separable,

(Suganthan

etal.,2005)

where

z=

x−

o=

fbias 6

=−450

andscalable

Shifted

f7(x

)=

N−1 ∑ i=1

(100(z

i+1−

z2 i)2

xi∈

min(f

7)

Multi-modal,shifted

Rosenbrock

+(z

i−

1)2)+

fbias 6,

[−100,1

00]

=f(o

1,...,o

N)

non-separable,

(Suganthan

etal.,2005)

where

z=

x−

o=

fbias 6

=−390

andscalable

Shifted

f8(x

)=

N ∑ i=1

(z2 i−

10cos(2πz i)+

10)

xi∈

min(f

8)

Multi-modal,shifted,

Rastrigin

+fbias 9,

[−5,5

]=

f(o

1,...,o

N)

separable,

(Suganthan

etal.,2005)

where

z=

x−

o=

fbias 9

=−330

andscalable


During the selection process the fitness of an individual should indicates thesurvaivability, and the reproduction of that individual (Back, 1996). A good selectionscheme strike the balance between the convergence reliability (exploration) and theconvergence velocity (exploitation) (Back, 1996).

Selection is the force that provide EA with the convergence level (Hancock, 1994).The selection scheme has to maintain the diversity of the population to avoid prematureconvergence (Miller and Goldberg, 1995). Selecting many solutions from the memorywill lead to a premature convergence, and selecting a small number of solutions frommemory will make the algorithm progresses slowly. A premature convergence is whengood individuals take over the population before reaching a global optimal solution(Miller and Goldberg, 1995).

Convergence rate is affected by the level of good individuals selected from thepopulation. The higher number of such individuals selected will result in a higherconvergence rate, but if this number is too high, a premature convergence could fastlyhappened. The takeover time is another measure that can be used to test the selectionpressure of a selection scheme. Selection pressure is the level of good individualselected from the population (Miller and Goldberg, 1995). Smaller takeover time meansstronger selection pressure (Back, 1994). Usually in early generations of EA most of theindividuals have weak fitness. A few good individuals are selected from the populationas individuals with strong fitness.

The takeover time is calculated by counting the number of iterations to reacha population with all its individuals, better than the best solution in the initialpopulation. Let P = {x1(1), x2(1), x3(1), . . . , xHMS(1)} be the set of the initialsolution at iteration one in the HM. Let xb(i) represents the best solution whereb = argminb∈[1,2,...,HMS] f(x

b(i)). The takeover time is equal to j, where j is theiteration number when all individuals in the HM have a fitness value similar orbetter than xb(1), such that f(x1(j)) ≤ f(x2(j)) ≤ f(x3(j)) ≤ . . . ≤ f(xHMS(j)) ≤f(xb(1)).

Figure 1 The trend of the best value for the benchmark functions (50,000 iterations),(a) sphere function (b) step function (c) camel-Back (d) Schwefel problem 2.26(e) shifted sphere (f) shifted Schwefel (g) shifted Rastrigin (h) shifted Rosenbrock(see online version for colours)

90

100

70

80

n

50

60

eneration

Random

40

50

est

Per Ge Random

Linear Rank

Scaled Proportional

20

30

Be

Tournament

0

10

0

1 10000 20000 30000 40000 50000

Generations

0000

nn

Randome Random

e

(a)

n

50

n

Random50

e Random

e

00

3

30

35

25

30

n

20

eneration

Random

15

est

Per Ge Random

Linear Rank

Scaled Proportional

10

Be

Tournament

5

0

1 10000 20000 30000 40000 50000

Generations

2 6060

1252012520

12480

n

12480

n

Randome Random

e

0000

nn

Randome Random

e

00

210210

30n 30

270

n

Random270

510

e Random

510

750

e

750

990990

000000

nn

Randome Random

e

(b)


Figure 1 The trend of the best value for the benchmark functions (50,000 iterations),(a) sphere function (b) step function (c) camel-Back (d) Schwefel problem 2.26(e) shifted sphere (f) shifted Schwefel (g) shifted Rastrigin (h) shifted Rosenbrock(continued) (see online version for colours)

0000

nn

Randome Random

e

(c)

2 60 12560

12520 12520

12480

n

12480

12440

eneration

Random

12400

est

!Per!Ge Random

Linear!Rank

Scaled Proportional

!12360

Be

Tournament

!12320

1 10000 20000 30000 40000 50000

Generations

0000

nn

Randome Random

e

00

210210

30n 30

270

n

Random270

510

e Random

510

750

e

750

990990

000000

nn

Randome Random

e

(d)

00 400

350

300

250

n

200

eneration

Random 150

100

est

!Per!Ge Random

Linear!Rank

Scaled!Proportional

50

0

Be

Tournament

50

100

1 3000 7000 11000

Generations

(e)

0000

nn

Randome Random

e

0 450

330

210 210

90

30n 30

150

270eneration

Random270

390

510est

!Per!Ge Random

Linear!Rank

Scaled!Proportional

510

630

750

Be

Tournament

750

870

990990

1 10000 20000 30000 40000 50000

Generations

000000

nn

Randome Random

e

(f)

(g)

000

4500

5000

4000

n

3000

3500

eneration

Random

2500

est

Per Ge Random

Linear Rank

Scaled Proportional

1500

2000Be

Tournament

1000

500

1 10000 20000 30000 40000 50000

Generations

(h)

A trend for the best individual in each generation is plotted in Figure 1. This figureshows the benchmark functions after applying four different selection schemes. Figure 2shows the takeover time is calculated in the implementation for 30 runs, by observingafter how many iterations the best individual in the initial population is copied. InFigure 3, the takeover time is averaged for the 30 runs to show the different selectionschemes rank according to the takeover time.


Figure 2 The takeover time on 30 runs, each with 50,000 iterations, (a) sphere function(b) step function (c) camel-Back (d) Schwefel problem 2.26 (e) shifted sphere(f) shifted Schwefel (g) shifted Rastrigin (h) shifted Rosenbrock(see online version for colours)

(a) (b)

(c) (d)

(e) (f)


Figure 2 The takeover time on 30 runs, each with 50,000 iterations, (a) sphere function(b) step function (c) camel-Back (d) Schwefel problem 2.26 (e) shifted sphere(f) shifted Schwefel (g) shifted Rastrigin (h) shifted Rosenbrock (continued)(see online version for colours)

(g) (h)

Figure 3 The average takeover time on 30 runs, each with 50,000 iterations,(a) sphere function (b) step function (c) camel-Back (d) Schwefel problem 2.26(e) shifted sphere (f) shifted Schwefel (g) shifted Rastrigin (h) shifted Rosenbrock(see online version for colours)

(a) (b)

(c) (d)


Figure 3 The average takeover time on 30 runs, each with 50,000 iterations,(a) sphere function (b) step function (c) camel-Back (d) Schwefel problem 2.26(e) shifted sphere (f) shifted Schwefel (g) shifted Rastrigin (h) shifted Rosenbrock(continued) (see online version for colours)

(e) (f)

(g) (h)

3.1 Design of experiments and standard test functions

Table 1 overviews a summary for eight global minimisation benchmark functions usedto evaluate different selection schemes most of which previously used in Omran andMahdavi (2008), Das et al. (2011), Pan et al. (2010) and Zou et al. (2010b). Thesebenchmark functions has been selected based on its characteristics as shown in the lastcolumn of Table 1. The benchmark functions were implemented with multi-dimensions(N = 30), with the exception to six-hump camel-back function which is two-dimensional.

We altered the memory consideration in HSA by incorporating the process of threeselection schemes: tournament, linear rank, and scaled proportional. Each one of themis considered as a variation of HSA. In order to standardise the evaluation process ofthe HSA variations we used the same parameter settings: HMCR = 0.94, PAR = 0.3,BW = 0.01, and NI = 50,000. These values are similar to what has been suggested in thestate of the art methods (Omran and Mahdavi, 2008; Das et al., 2011; Pan et al., 2010;Zou et al., 2010b). We selected the HMS = 20 to see the effect of different selectionschemes on the performance of HSA. Notice that the size of the HM chosen does notsignificantly affect the performance of the HSA (Al-Betar et al., 2011).

For tournament selection, the tournament size selected is 2. For the linear rankselection, η+ = 1.1, this value is recommended (Back, 1994; Blickle and Thiele, 1997).


For proportional selection the scaling value c = 2, this value is recommended (Mitchell,1996; Eiben and Smith, 2003).

3.2 Random selection

This selection scheme is the standard method used in the original HSA presented byGeem et al. (2001). In this selection, the memory consideration selects the value ofthe decision variable in the new harmony randomly (i.e., uniform selection) from anysolution in the HM.

Firstly a random solution xi is selected from HM, then the corresponding value ofthe decision variable x′

j is selected, x′j = xi

j .Figure 1 summarises the experimental results show that the random selection has

mostly the worst convergence rate for all the benchmark functions compared with otherselection schemes, except in the case of six-hump camel-back since the random selectionis inferior with problem dimension. The random selection scheme has mostly the highesttakeover time as shown in Figure 3.

3.3 Tournament selection

The tournament selection scheme chooses t number of solutions from the HM(Goldberg et al., 1989; Motoki, 2002). The best solution from this tournament isselected as a candidate for the next generation population. In our experiment we usedt = 2 (binary tournament) which is the suggested tournament size (Blickle and Thiele,1997).

Figure 1 shows that binary tournament selection has the second best convergencerate for all the benchmark functions, except in the case of the Shifted Rosenbrockfunction which has the worst convergence rate. It does not give good results for aproblem with multi-modal, non-separable characteristic. The binary tournament selectionhas the second lowest takeover time as Figure 3 shows.

3.4 Scaled proportional selection

The proportional selection scheme is the first selection developed for GA (Holland,1975). During this selection scheme the values of each decision variable is selected fromthe solutions stored in HM, based on a probability associated with each solution.

The proportional selection selects individuals according to their fitness values withrespect to the fitness of other individuals in the population (Holland, 1975; Goldberg,1989) i.e.,

pi =f(xi)

HMS∑j=1

f(xj)

where xi ∈ HM (2)

The proportional selection scheme is un-scaled. This means that negative fitness valuesare not applicable using this selection scheme, and it will work only if all fitness valuesare positive (Blickle and Thiele, 1997). The un-scaled proportional selection can not be


applied in the case of the possibility of negative values of the objective function (Back,1996).

In our previous work (Al-Betar et al., 2011), we used un-scaled proportionalselection scheme with HSA. The experiments show that the results of this selectionscheme is far from the optimal solution in the case of benchmark functions with negativefitness value. In order to overcome this problem we used scaled proportional selectionscheme.

In our application of the proportional selection scheme we used sigma scaling. It isa fitness re-mapping scheme that can be used when there is a possible negative valuesin the objective function (Mitchell, 1996; Eiben and Smith, 2003) as follows:

f ′(xi) = max (f(xi)− (M − c× σ), 0.0) (3)

where c is a small integer usually set to the value 2, σ is the population fitness standarddeviation, M is the population fitness mean, f(xi) is the fitness of individual i in theHM, and f ′(xi) is the scaled fitness.

Figure 1 shows that scaled proportional selection has the worst convergence rate forthe benchmark functions. This is because during early stages of the HSA a small numberof individuals with a similar fitness are converged quickly. This leads the HSA to betrapped by a local minima. The scaled proportional selection has the highest takeovertime as Figure 3 shows.

3.5 Linear rank selection

The linear rank selection scheme was proposed by Baker (1985), to overcome theproblems found in the proportional selection (Whitley, 1989). For the linear rankselection, we first sort the individuals in the HM according to their fitness value fromworst to best. The probability of choosing an individual is assigned using the equation(Al-Betar et al., 2011):

pi =1

HMS×

(η+ − (η+ − η−)× i− 1

HMS− 1

)(4)

where i is the rank index of the solution xi, ∀i ∈ (1, 2, . . . ,HMS). The assumption:∑HMSi=1 pi = 1 and pi ≥ 0,∀i ∈ (1, 2, . . . ,HMS).This selection scheme require that 1 ≤ η+ ≤ 2 and η+ = 2− η− be fulfilled.

Normally, the value of η+ determines the selective pressure which is determined inadvance and η+ = 1.1 is recommended (Back, 1994).

Figure 1 shows that linear rank selection has the fastest convergence rate for thebenchmark functions, as this selection scheme depends on the rank of the individualwithin the HM. This selection method selects few good individuals in each stage asthis would result in a premature convergence. The linear rank selection has the lowesttakeover time as shown in Figure 3.

4 Comparative analysis

The takeover time and convergence rate are important properties of a selection scheme,which impact the optimisation trend and the solution obtained by HSA. The results


show that the optimisation task and the type of the problem to be solved affects theconvergence speed and the takeover time of the selection scheme.

Figure 1 shows that linear rank selection has the fastest convergence. Thisselection scheme depends on the individual rank regardless of the relative fitness. Thisresult in selecting fewer good individuals, and minimise the probability of prematureconvergence.

Figure 1 presents clearly that random selection leads to the slowest convergencefor the shifted Rastrigin. The multi-modal, shifted, and separable characteristics ofthis function makes the random selection of the solutions not that effective in solvingproblems with such characteristics.

On the other hand, the results show clearly that binary tournament selection has theslowest convergence for the Shifted Rosenbrock. The selection of local best solutionfrom two randomly selected solutions does not give enough diversity for problems withsuch characteristic.

For all other six functions (i.e., sphere, step, camel-back, Schwefel problem 2.26,shifted sphere, and shifted Schwefel) the scaled proportional selection has the slowestconvergence. This selection scheme relies on the individual fitness value to be choosed,which result usually in choosing individuals with high fitness in early stages of thesearch. This result usually in a premature convergence.

Figure 2 shows that the tournament selection scheme has the lowest takeover timefor the Shifted Rosenbrock function. While the lowest takeover time for most of otherfunctions is observed in the linear rank selection. Observing Figure 3, it can be seenclearly that random and scaled proportional selection have the highest average takeovertime for the benchmark functions.

On the other hand, linear rank selection has the lowest average takeover timefor the benchmark functions. The linear rank selection scheme has a high degree ofextinctivness, which result in a lower diversity in the HM. As partial solutions are notmaintained to be used for further improvements in the next generations.

5 Conclusions

We give a unified and systematic analysis of common selection schemes used in memoryconsideration operation of the HSA. The selection schemes are: random, tournament,linear rank, and scaled proportional. These selection schemes is analysed using eightglobal benchmark functions commonly used in the literature in terms of takeover timeand convergence rate.

In order to improve the performance of HSA, selection schemes that apply thenatural selection principle substituted the random selection scheme. This can help inguiding the process of searching for the optimal solution.

The experimental results show that the linear rank selection has the lowest takeovertime, and it has the fastest convergence. The results show that scaled proportionalselection has the highest takeover time and the lowest convergence speed.

The binary tournament result in slow convergence for functions with specificcharacteristics (e.g., multi-modal, shifted, and non-separable). The selection of localbest solution from two randomly selected solutions does not give enough diversity forproblems of such characteristics.


The scaled proportional selection give weak results for the benchmark functions. Inthis selection scheme the probability of selecting an individual depends on its fitnessvalue. This result in choosing individuals with similar fitness in early stages of thesearch, and this leads to a premature convergence.

The future work can be experimenting different selection schemes parameters(i.e., t in the tournament selection, η+ in the linear rank, c in the scaled proportionalselection schemes).

ReferencesAl-Betar, M.A. and Khader, A.T. (2010) ‘A harmony search algorithm for university course

timetabling’, Annals of Operation Research, Vol. 194, No. 1, pp.3–31.Al-Betar, M.A., Khader, A.T. and Thomas, J.J. (2010a) ‘A combination of metaheuristic

components based on harmony search for the uncapacitated examination timetabling’,in 8th International Conference on the Practice and Theory of Automated Timetabling(PATAT 2010), Belfast, Northern Ireland.

Al-Betar, M.A., Khader, A.T. and Liao, I.Y. (2010b) ‘A harmony search algorithm withmulti-pitch adjusting rate for university course timetabling’, in Z.W. Geem (Ed.): RecentAdvances in Harmony Search Algorithm, Vol. 270 of SCI, pp.147–162, Springer-Verlag,Berlin, Heidelberg.

Al-Betar, M.A., Khader, A.T. and Nadi, F. (2010c) ‘Selection mechanisms in memoryconsideration for examination timetabling with harmony search’, in GECCO ‘10, Portland,Oregon, USA.

Al-Betar, M.A., Doush, I.A., Khader, A.T. and Awadallah, M.A. (2011) ‘Novel selectionschemes for harmony search’, Applied Mathematics and Computation, Vol. 218, No. 10,pp.6095–6117.

Back, T., Hoffmeister, F. and Schwefel, H-P. (1991) ‘Extended selection mechanisms in geneticalgorithms’, in R. Belew and L. Booker (Eds.): Proceedings of the Fourth InternationalConference on Genetic Algorithms, pp.92–99, Morgan Kaufmann, San Mateo.

Back, T. (1994) ‘Selective pressure in evolutionary algorithms: A characterization of selectionmechanisms’, in Proceedings of the First IEEE Conference on Evolutionary Computation,IEEE Press, pp.57–62.

Back, T.(1996) Evolutionary Algorithms in Theory and Practice, Oxford University Press,New York, Oxford.

Baker, J.E. (1985) ‘Adaptive selection methods for genetic algorithms’, in L. Erlbaum (Ed.):Proceedings of the 1st International Conference on Genetic Algorithms, pp.101–111,Associates Inc., Hillsdale, NJ, USA.

Blickle, T. and Thiele, L. (1997) ‘A comparison of selection schemes used in evolutionaryalgorithms’, Evolutionary Computation, Vol. 4, No. 4, pp.361–394.

Das, S., Mukhopadhyay, A., Roy, A., Abraham, A. and Panigrahi, B.K. (2011) ‘Exploratorypower of the harmony search algorithm: analysis and improvements for global numericaloptimization’, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics,Vol. 41, No. 1, pp.89–106.

Eiben, A.E. and Smith, J.E. (2003) Introduction to Evolutionary Computing, SpringerVerlag,Natural Computing Series.

Geem, Z.W., Kim, J.H. and Loganathan, G.V. (2001) ‘A new heuristic optimization algorithm:harmony search’, Simulation, Vol. 76, No. 2, pp.60–68.

Geem, Z.W., Kim, J.H. and Loganathan, G.V. (2001) ‘A new heuristic optimization algorithm:harmony search’, Simulation, Vol. 76, No. 2, pp.60–68.


Geem, Z.W. (2008a) ‘Harmony search applications in industry’, Soft Computing Applications inIndustry: Studies in Fuzziness and Soft Computing, Vol. 226, pp.117–134.

Geem, Z.W. (2008b) ‘Novel derivative of harmony search algorithm for discrete designvariables’, Applied Mathematics and Computation, Vol. 199, No. 1, pp.223–230.

Geem, Z.W. (2009) ‘Particle-swarm harmony search for water network design’, EngineeringOptimization, Vol. 41, No. 4, pp.297–311.

Goldberg, D. and Deb, K. (1991) ‘A comparative analysis of selection schemes used in geneticalgorithms’, in G.J.E. Rawlins (Ed.): Foundations of Genetic Algorithms, pp.69–93, MorganKaufmann, San Francisco, CA.

Goldberg, D., Deb, K. and Korb, B. (1989) ‘Messy genetic algorithms: motivation, analysis, andfirst results’, Complex Systems, Vol. 3, pp.493–530.

Goldberg, D.E. (1989) Genetic Algorithms in Search, Optimization and Machine Learning,Addison Wesley.

Hancock, P.J.B. (1994) ‘An empirical comparison of selection methods in evolutionaryalgorithms’, in Selected papers from AISB Workshop on Evolutionary Computing,Springer-Verlag, London, UK, pp.80–94.

Holland, J.H. (1975) Adaptation in Natural and Artificial Systems, The University of MichiganPress, Ann Arbor, MI, USA.

Ingram, G. and Zhang, T. (2009) ‘Overview of applications and developments in the harmonysearch algorithm’, in Z.W. Geem (Ed.): Music-Inspired Harmony Search Algorithm, Vol. 191of SCI, pp.15–37, Springer-Verlag, Berlin, Heidelberg.

Lee, K.S. and Geem, Z.W. (2005) ‘A new meta-heuristic algorithm for continuous engineeringoptimization: harmony search theory and practice’, Computer Methods in Applied Mechanicsand Engineering, Vol. 194, Nos. 36–38, pp.3902–3933.

Mahdavi, M., Fesanghary, M. and Damangir, E. (2007) ‘An improved harmony search algorithmfor solving optimization problems’, Applied Mathematics and Computation, Vol. 188, No. 2,pp.1567–1579.

Miller, B. and Goldberg, D. (1995) Genetic Algorithms, Selection Schemes, and the VaryingEffects of Noise, Illegal Report 95009, University of Illinois at Urbana-Champaign.

Mitchell, M. (1996) An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA.Motoki, T. (2002) ‘Calculating the expected loss of diversity of selections schemes’, Evolutionary

Computation, Vol. 10, No. 4, pp.397–422.Omran, M.G.H. and Mahdavi, M. (2008) ‘Global-best harmony search’, Applied Mathematics

and Computation, Vol. 198, No. 2, pp.643–656.Pan, Q-K., Suganthan, P., Tasgetiren, M.F. and Liang, J. (2010) ‘A self-adaptive global best

harmony search algorithm for continuous optimization problems’, Applied Mathematics andComputation, Vol. 216, No. 3, pp.830–848.

Pan, Q-K., Suganthan, P., Liang, J. and Tasgetiren, M.F. (2011) ‘A local-best harmony searchalgorithm with dynamic sub-harmony memories for lot-streaming flow shop schedulingproblem’, Expert Systems with Applications, Vol. 38, No. 4, pp.3252–3259.

Pandi, V.R. and Panigrahi, B.K. (2011) ‘Dynamic economic load dispatch using hybrid swarmintelligence based harmony search algorithm’, Expert Systems with Applications, July, Vol.38, No. 7, pp.8509–8514.

Suganthan, P.N., Hansen, N., Liang, J.J., Deb, K., Chen, Y-P., Auger, A. and Tiwari, S.(2005) Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session onReal-parameter Optimization, Technical Report KanGAL Report #2005005, IIT Kanpur,India, Nanyang Technological University, Singapore.


Taleizadeh, A.A., Niaki, S.T.A. and Barzinpour, F. (2011) ‘Multiple-buyer multiple-vendormulti-product multi-constraint supply chain problem with stochastic demand and variablelead-time: a harmony search algorithm’, Applied Mathematics and Computation, Vol. 217,No. 22, pp.9234–9253.

Wang, C-M. and Huang, Y-F. (2010) ‘Self-adaptive harmony search algorithm for optimization’,Expert Syst. Appl., April, Vol. 37, No. 4, pp.2826–2837.

Wang, L., Pan, Q-K. and Tasgetiren, M.F. (2010) ‘Minimizing the total flow time in a flowshop with blocking by using hybrid harmony search algorithms’, Expert Systems withApplications, Vol. 37, No. 12, pp.7929–7936.

Wang, L., Pan, Q-K. and Tasgetiren, M.F. (2011) ‘A hybrid harmony search algorithm for theblocking permutation flow shop scheduling problem’, Computers & Industrial Engineering,Vol. 61, No. 1, pp.76–83.

Whitley, D. (1989) ‘The genitor algorithm and selection pressure why rank based allocation ofreproductive trials is best’, in Proceedings of the Third International Conference on GeneticAlgorithms, Morgan Kaufmann, pp.116–121.

Yao, X., Liu, Y. and Lin, G. (1999) ‘Evolutionary programming made faster’, IEEE Transactionson Evolutionary Computation, Vol. 3, No. 2, pp.82–102.

Zhao, S-Z., Suganthan, P., Pan, Q-K. and Tasgetiren, M.F. (2011) ‘Dynamic multi-swarmparticle swarm optimizer with harmony search’, Expert Syst. Appl., April, Vol. 38, No. 4,pp.3735–3742.

Zou, D., Gao, L., Li, S. and Wua, J. (2010a) ‘An effective global harmony search algorithm forreliability problems’, Expert Systems with Applications, Vol. 38, pp.4642–4648.

Zou, D., Gao, L., Wu, J. and Li, S. (2010b) ‘Novel global harmony search algorithm forunconstrained problems’, Neurocomputing, Vol. 73, Nos. 16–18, pp.3308–3318.

analysisoftakeovertimeandconvergencerate … bany mohammed is an assistant professor at the mis...

Documents